Supervisory Measurement-Guided Noise Covariance Estimation: Discussing Forward and Reverse Differentiation

Reliable state estimation depends on accurately modeled noise covariances, which are difficult to determine in practice. This paper formulates the noise covariance estimation as a bilevel optimization problem that factorizes the joint likelihood of primary and supervisory measurements to reconcile information exploitation with computational tractability. The factorization converts the nested Bayesian dependency into a Markov-chain structure, allowing efficient computation. At the lower level, a Kalman filter with state augmentation performs such computation. Meanwhile, closed-form forward and reverse differentiation provide efficient gradients for the upper-level updates, and we compare the two models' space and time complexities to inform their practical selection. The upper level subsequently refines the noise covariances to guide the lower-level estimation. Taken together, the proposed algorithms offer a systematic and computationally efficient approach to noise covariance estimation in linear Gaussian systems.